Question 167: Centre of Mass and Rotational Motion

Question

A wedge of mass M = 4m lies on a frictionless
plane. A particle of mass m approaches the
wedge with speed v. There is no friction
between the particle and the plane or between
the particle and the wedge. The maximum
height climbed by the particle on the wedge is
given by :-

Accepted Answer

Initial condition can be shown in the figure
below
As mass $m$ collides with wedge, let both wedge and mass move with speed $v^{\prime}$. Then,
Given:
Mass of the wedge (M) = 4m
Mass of the particle (m)
Initial speed of the particle (v)
There is no friction.
Step 1 : Conservation of Linear Momentum
Before the collision, the momentum of the system is just the momentum of the particle because the wedge is at rest. After the collision, both the particle and the wedge will be moving. Let's denote the final common velocity as $v'$.
We can write the conservation of momentum as :
$$mv = (m + 4m)v'$$
which simplifies to
$$v' = \frac{v}{5} 	ag{1}$$
Step 2 : Conservation of Mechanical Energy
We apply the conservation of energy before and after the collision. Before the collision, only the particle has kinetic energy. After the collision, both the particle and the wedge have kinetic energy and the particle has potential energy due to its height h on the wedge.
We can write the conservation of energy as :
$$\frac{1}{2}mv^2 = \frac{1}{2}(m + 4m){v'}^2 + mgh$$
which simplifies to
$$mv^2 = (m + 4m){v'}^2 + 2mgh$$
Substituting equation (1) into this equation gives
$$v^2 = 5{v'}^2 + 2gh \Rightarrow \frac{4}{5}v^2 = 2gh$$
which simplifies to
$$h = \frac{2v^2}{5g}$$
So, the maximum height climbed by the particle on the wedge is given by $\frac{2v^2}{5g}$.
Therefore, the correct answer is Option D :
$$\frac{2v^2}{5g}$$

Answer

$${{{v^2}} \over {g}}$$

Answer

$${{2{v^2}} \over {7g}}$$

Answer

$${{{v^2}} \over {2g}}$$

Answer

$${{2{v^2}} \over {5g}}$$

Centre of Mass and Rotational Motion

A wedge of mass M = 4m lies on a frictionless plane. A particle of mass m approaches the wedge with speed v. There is no friction between the particle and the plane or between the particle and the wedge. The maximum height climbed by the particle on the wedge is given by :-

Choose the correct answer: